{
  "abstract": "We study uniform high-frequency localization for the Laplacian on compact metric graphs through the least $L^2$-mass that eigenfunctions must place in a prescribed measurable observation set. We first identify this asymptotic localization constant with the minimum of a linear functional over the attainable edge-intensity set; in the generic standard-Kirchhoff setting, this set is governed by the regular Gauss image of the secular manifold. Primitive cycles and exterior-to-exterior paths consequently determine the positivity threshold, but not, in general, the positive numerical value. We introduce a boundary-aware singular-completion cone and prove that it contains all regular secular edge-energy vectors for trees, unicyclic graphs, and closed graphs of cycle rank two. We then construct a cycle-rank-two graph with two Dirichlet leaves for which this completion principle fails. An exact rational separator, combined with a validated Krawczyk enclosure, yields a nonsingular scalar secular state lying outside every boundary-compatible singular sector. A positive radial derivative identity and recurrence in the compact orbit closure convert this local separation into an exact high-frequency eigensequence for a single fixed metric. For a suitable measurable observation set, the true high-frequency localization constant $C_\\infty(ω;\\ell)$ and its singular-completion counterpart $C_{\\mathrm{sing}}(ω;\\ell)$ satisfy $$C_\\infty(ω;\\ell)<1/2<C_{\\mathrm{sing}}(ω;\\ell)$$. Thus, singular completion captures the quantitative localization geometry in several low-complexity classes but does not, in general, determine the high-frequency variational problem on a fixed quantum graph.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2609.17623",
  "author_total": 1,
  "authors": [
    {
      "name": "Binh T. Nguyen"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/55861yp9d9p3b.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2609.17623",
  "fulltext_url": null,
  "id": "55861yp9d9p3b",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2609.17623",
  "language": null,
  "licence": "CC-BY-NC-ND-4.0",
  "licence_label": "CC BY-NC-ND 4.0",
  "licence_source": "article",
  "licence_url": "http://creativecommons.org/licenses/by-nc-nd/4.0/",
  "metadata_licence": "CC0-1.0",
  "native_subjects": [
    [
      "arxiv",
      "math-ph"
    ],
    [
      "arxiv",
      "math.MP"
    ],
    [
      "arxiv",
      "math.SP"
    ],
    [
      "openalex",
      "2610|Spectral Theory in Mathematical Physics"
    ]
  ],
  "openalex": "W7213500836",
  "pdf_url": "https://arxiv.org/pdf/2609.17623",
  "pmcid": null,
  "pmid": null,
  "published": "2026-09-15",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/55861yp9d9p3b.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:22:01Z",
      "native_id": "2609.17623",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2609.17623"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.SP",
      "subfield": "analysis-pdes"
    },
    {
      "field": "mathematics",
      "origin": "arxiv:math-ph",
      "subfield": "mathematical-physics"
    }
  ],
  "title": "Uniform High-Frequency Localization on Quantum Graphs",
  "title_text": "Uniform High-Frequency Localization on Quantum Graphs",
  "type": "preprint",
  "updated_at": "2026-10-30T12:22:04Z",
  "url": "https://research.draconitic.com/paper/55861yp9d9p3b",
  "venue": "arXiv",
  "versions": []
}